Numerical simulations of piano strings. II. Comparisons
with measurements and systematic exploration of some
hammer-string parameters
AntoineChaigne
SignalDepartment,TelecomParis;CNRS U1L4820, 46 rue Barrault, 75634ParisCedex13, France
Anders Askenfelt
Departmentof SpeechCommunication
and Music.4coustics,
RoyalInstituteof Technology
(KTH),
P.O. Box 700 14, S-100 44 Stockhoi'm, Sweden
(Received26 May 1993;acceptedfor publication30 November1993)
A physicalmodel of the piano string, usingfinite differencemethods,has recentlybeen
developed.
[ChaigneandAskenfelt,J. Acoust.Soc.Am. 95, 1112-1118( 1994)].The modelis
basedon the fundamentalequationsof a damped,stiff stringinteractingwith a nonlinear
hammer,from whicha numericalfinitedifference
schemeis derived.In the presentstudy,the
performanceof the model is evaluatedby systematiccomparisons
betweenmeasuredand
simulatedpianotones.After a w•rificationof the accuracyof the method,the modelis usedas
a toolfor systematically
exploringthe influence
of stringstiffness,
relativestrikingposition,and
hammer-string
massratio on stri.ngwaveforms
and spectra.
PACS numbers: 43.75.Mn, 43.75.Wx
INTRODUCTION
In a previouspaper, a physicalmodel of the piano
stringwithparticularemphasis
ona time-domain
modeling
closeto the basicequations
hasbeenpresented.
• The
model,whichis basedon the one-dimensional
equations
of
a damped,stiffstringinteractingwith a nonlinearhammer,
is put into a numericalform by the useof standardfinite
differencemethods.The compression
characteristicof the
hammer,beingone of the most essentialcomponents,
is
modeledby a powerlaw. Usingthismodel,principalquantitieslike stringdisplacement
and velocity,the interacting
forcebetweenhammerand string,and the'.forceexertedby
the stringon the bridge,can convenientlybe computedfor
differentcombinations
of hammerand stringparameters.
The modelhasnowbeenevaluatedfor a largenumber
of casesby comparingsimulatedwaveformswith measurementson real pianos.The comparisons
iincludeda set of
representative
notesof the pianofrom the bass,mid, and
trebleregister,playedat differentdynamiclevels,and with
differenthammers.The hammer-string
p•trameters
usedin
usinga modelwhichhasbeenverifiedto generaterealistic
results.The parametersincludedin the presentpart II of
the study are relative striking position, hammer-string
massratio, and stringstiffness,
whichall attracta particular interestsincetheir exact valuesappearto be crucial
for piano design.
I. COMPARISON
PLANO STRING
BETWEEN REAL AND SIMULATED
VIBRATIONS
In this section,the numericalmodel is evaluatedby
comparingsimulatedstring and hammer waveformswith
previously
measured
signals
in realpianos?
'• Otherexamplesof comparisons
betweenreal and simulatedwaveforms
can be foundin previousworkson pianostringsbasedon
different
numerical
techniques.
6'7
The numericalvaluesof all physicalparametersused
in the presentexperimentsare givenin Table I. Someof the
parameterswere measuredby the authors, while others
wereextracted
fromtheliterature.
3'5'8
Theparameters
of
the hammerfelt (stiffness
coefficient
K, nonlinearityexpo-
the simulations were obtained from measurements.
nentp), andthedamping
coefficients
ofthestring(bI ,b3)•
After the methodhadbeenverifiedto reproduce
measuredwaveformssatisfactorily,the model was usedfor a
systematicexplorationof someof the hammer-stringparametersandtheirinfluenceon stringwaveforms
andspectra. The underlyingtheory of the struckstring and the
principaldependence
of the hammer-string
parametersis
were estimatedfrom experimentaldata by meansof standard curve-fittingprocedures.
nowwellknown,
2'3butinmanycases
anexperimental
verificationhasbeenlackingdueto the practicaldifficulties
of
reachinga desiredstep-by-step
variationin only one pa-
A. Bass, mid, and treble
A firstcheckof the modelwasmadeby computingthe
stringwaveformsfor a few notesin differentrangesof the
piano.In orderto clearlyillustratethe propagationon both
sides of the hammer, it was decided to observethe veloci-
tiesat somedistancefrom the strikingpositionratherthan
at that point. Figure 1 showsa comparisonbetweenthe
is particularlywell suited,and severalinvestigators
have simulatedand observedstringvelocitiesat 40 mm from the
tried this approach,usingmodelsof wiryingdegreeof striking point towards the bridge, for a treble (C7),
refinement.•
Thisstudy
addsa moresystematic
approach, midrange(C4), andbassnote (C2), respectively.
It canbe
rameter at a time. For this purpose,a numericalsimulation
1631 J. Acoust.Soc.Am.95 (3), March1994
0001-4966/94/95(3)/1631/10/$6.00
@ 1994Acoustical
Societyof America 1631
TABLE I. Valuesof the parametersusedin the simulations.
Parameters
C2
C4
C7
Units
65.4
1.90
35.0(X2)
750
262
0.62
3.93(X3)
670
2093
0.09
0.467(• 3)
750
Hz
m
g
N
2.0X10TM
3.82X10-s
0.5
6.25• 10-9
2.0X1011
8.67X10-4
0.5
2.6X 10-•0
St•ing
fundamentalfrequency
stringlength
total stringmass
stringtension
f•
L
Ms
T
Young's
modulus
stringstiffness
parameter
damping
coefficient
E
ß
b•
damping
coefficient
b3
2.0X1011
7.5X10-6
0.5
6.25X 10-9
a
Mx
p
K
Mx/M s
O.12
4.9 (X2)
2.3
4.0• l0 s
0.14
O.12
2.97 (X3)
2.5
4.5• 109
0.75
0.0625
2.2 (>(3)
3.0
1.0• 1012
4.71
f•
N
16
100
32
50
96
16
N/m2
s- •
s
Hammer
relativestrikingposition
hammermass
stiffnessnonlinearexponent
coefficient
of hammerstiffness
hammer-string
massratio
g
Sampling
samplingfrequency
numberof stringsegments
seenthat the model is able to reproducethe waveforms
quite well over the whole registerof the piano, including
the details of the attack transients.Small discrepancies
which can be observedin the actual timing relations between the pulsesare mostlydue to slight differences
in
observationpoints.In the C2-example,the agrafiereflection during stringcontact(the first negativepulse) is not
reproducedaccurately.This discrepancyis likely due to
either a nonrigidstring terminationat the agrafiein the
real piano,or an interferencewith reflectionsfrom the end
of the string wrapping. These two phenomenaare not
taken into account in the model.
1.6 ms/o•v
kHz
The computedforcesbetweenhammerand stringfor
the C7, C4, and C2 examplesaboveare shown in Fig. 2.
This variable is of particular interestas it determinesthe
stringspectra?
Theactualhistory
of thehammer
forceis
difficultto measure
undernormalconditions,
mandanindirect estimationby a measurementof the accelerationat
the woodenhammer mouldingis normally the closestalternative. Such measurements, which have been exten-
sively investigatedin a previouswork by the secondau-
thor,arenotreported
in thepresent
paper.
n
The simulationsemphasizethe strongeffectof the reflectionsfrom the agrafie during hammer string contact,
clearly seen in the C4 and C2 examples.Each reflection
resultsin a temporaryincreasein interactingforce, correspondingto a downwardimpulseon the hammer.As dis-
1.6 ms/olv
0
I
2
3
ms]
O,
uJ
)
m/s
N15]
0
z
•-
-2.0
1.6 ms/olv
•
0
1
2
3
m9
20
0
-
N
1o
o
o
i
2
•
ms
FIG. 1. Stringvelocityat the bridgesideof the strikingpoint (40 mm
from the hammer) for a bass,mid, and treble note (C2-C4-C7), played
to Fig. 1. Initial hammezzoforte:
measured
4 (fullline)andsimulated
(dashed).
Themeasured FIG. 2. Simulatedhammerforcescorresponding
hammer-string
contactdurationis includedfor reference.
1632 d. Acoust.Soc. Am., Vol. 95, No. 3, March 1994
mer velocity Vno=2.5 m/s.
A. Chaigneand A. Askenfelt:Simulationsof piano strings 1632
30
1.0
N
20
10
0
1
ms
2
I
ms
2
a
!
m•
a
0
I
ma
2
0
I
ms
2
o0
I
ms
2
-1.0
1.0
30
ß•
lO
•
-1.0
o
0
1.0
l
. --
•0
N
0
20
10
-1.0
o •
(a)
FIG. 3. Comparisonof string velocitiesat three dynamiclevels,fortemezzoforte-piano.
(C4, observation
pointat the bridge:side,40 mm from
thehammer).Measured
waveform(full line),s andsitmulated
(dashed)
with initialhammervelocity(l/•m) equalto 4.0, 1.5,and0.5 m/s.
cussedby Hall, thesereflectionsare oneof the major mechanismsof hammer release?
The peak forcein theseexamplesat mezzoforte level
were about40, 13, and 16 N, for stringsC7, C4, and C2,
respectively.The high value for the C7 exampleis due to
the pronounced
nonlineartryof the treblehammers(in this
casep=3.0). The hammer-stringcontact durations obtained in the simulationswere 0.6 ms (C7), 1.9 ms (C4)
and 3.25 ms (C2), respectively,
which is in closeagreement (within 6%) with the experimentally
observedvalues (0.6; 2.0; 3.1 ms).
B. Dynamic level
One of the mostimportantcontrolparameters
for the
pianoplayeristheinitialhammervelocity(V•). By varying this parameterin our nonlinearphysicalmodel,both
the level and frequencycontentof the simulatedtoneswill
change,like in real pianos.This ability of the model is
illustratedin Fig. 3 whichshowsa comparison
of measured
and simulatedwaveformsof the stringvelocityfor noteC4
playedat three differentlevels:forte, rnezzo
forte, and piano. The plots are presentedon a normaliizedamplitude
scale.Like in Fig. 1, the velocitieswereobservedat 40 mm
from the hammeron the bridgeside.In thc synthesis,
the
initial hammer velocitieswere 4.0, 1.5, and 0.5 m/s, respectively.
Again, the simulationsreproducethe rnainfeaturesof
themeasured
waveforms
convincingly,
includingthe steepening of the initial slopeand generalshorteningof the
pulseswith increasingdynamiclevel. Also, the contribution from stringstiffness,
clearlyvisiblein theforte example as a "ripple" with increasingamplitudeat the end of
the first period,is nicely reproducedin the simulations(see
also Sec. II A). The major differencebetweenmeasure-
mentsand simulationsis found at the first agrafiereflectionsin theforte examplewherea completematchwasnot
1633 J. Acoust.Soc.Am.,Vol.95, No. 3, March'1994
(b)
3
FIG. 4. Simulatedhammer forcesfor note C4 at three dynamic levels
(f-mr-p) corresponding
to Fig. 3; (a) absolutevaluesin N; (b) normalized.
reached,possiblyindicatinga slight discrepancyin the,
hammer parametersbetweenthe real and simulatedcase.
The calculatedhammer forcescorresponding
to the
threeexamplesatforte, mezzoforte,andpianolevelsin Fig..
3 areshownin Fig. 4(a). The simulations
show,in partic-.
ular,howthenonlinearstiffness
narrowstheforcepulsesas
the initial hammer velocity is increased.The peak force.,
increasesfrom about 2 N at piano to 22 N at forte. The',
normalized plots in Fig. 4(b) illustrate clearly how the',
waveformof the hammer-stringforcesharpensas the dy-.
namiclevelis increased.Also the effectof the agrafiere-.
flectionsbecomesmore prominent.
As expected,the spectraof the string velocityfor the,.
three casesshow an increasingspectral content with in-.
creasinghammervelocity(seeFig. 5). A comparisonbe-.
tween the measuredand simulatedspectrarevealsthat:
there are rather large differences
abovethe first 5-7 parA. ChaigneandA. Askenfelt:
Simulations
of pianostrings 1633
6O
aøl
0
"'
2
4
6
S
10
kHz
c4•
•c•
o0I
2
?
4
6
8
10
kHz
60
45
C
3O
15
0
0
2
4
6
8 kHzlO
FIG. 6. Comparisonof stringvelocitiesfor note C4 playedwith a treble
(C/H), original(C4H) and a basshammer(C2H); as measured(full
lines),
4andsimulated
(dashed).
Approximate
scale
unitsareI m/s.The
FIG. $. Spectralenvelopescorresponding
to the initial 32 ms of the
simulatedstring velocitiesat three dynamiclevels (timf-p) in Fig. 3
(dashedlines). The measuredspectralenvelopesare includedfor com-
measuredhammer-stringcontactdurationsare includedfor reference.
parison
(full lines).
•
experimentson real pianos,the hammer parameterswere
changedboth in a straightforward
mannerby exchanging
tials,in spiteof the seemingly
closematchin waveformin
Fig. 3. In particular,theforte exampleshowslocaldiscrepanciesof the order of 10-15 dB, which for the higher
modes,however,can be partiallyattributedto the limited
spatialresolutionof the numericalmodeling.Noticingthat
the maximumdeviationbetweenexact and approximate
locationof the striking point is equal to half the spatial
step,then the spatialresolutioncan be written L/2N (see
Table I for the definitionof variablesand symbols).As a
consequence,
the partials'frequencies
will not be correct,
and this error will in turn influencethe spectralenvelope.
An estimatefor the frequencyshift of the nth-partialcon-
hammers,
andalsoin thetraditional
waybyvoicing.
5'8
A comparisonof real and synthesizedwaveformsfor
the noteC4 playedwith exchangedhammers(C7, CA, and
C2 hammer) is givenin Fig. 6. The main differences
betweenthe threehammersare foundin the stiffness(p and
K), increasing
frombassto treble,but alsothe massdiffers
(seeTable I). Again, a convincingmatch in waveformis
seen,possiblywith a somewhatmorepeakedappearance
of
the simulatedwaveforms.The correspondingspectrain
Fig. 7 showa similaramountof high-frequency
contentfor
the measuredand simulatedcases,respectively.Like in
Fig. 5, a frequencyshift in the modeledspectralenvelope
secutive to a difference between real and modeled hammer
can be observed, relative to the measured, whose order of
locationcan be found by writing that the productof the
wave numberby the distanceof hammer from agrafieis
constant,which is equivalentto
magnitudeis givenby Eq. (1). In line with expectations,
the spectragrow richer with increasinghammerstiffness.
The hammerforcesfor the threehammer-string
combinationsare shownin Fig. 8, whereit can be seenhow the
smoothness
of the forcepulsesis changedby shiftingham-
1
Af, =f"'l+2Ns'
(1)
mers. With the soft, and more massivebass hammer, the
agrafiereflectionsare almostfilteredout. Also the contact
durationchangesslightly,increasingwith the softerand
heavierbasshammer,and decreasingwith the harderand
lighter treble hammer.However,thesechangesare relatively smal, about 0.1 ms (+5%). The changesare
smaller than those observed experimentally (about
4-15%), possiblyindicatinga smallerror in the estimation
of the hammerparameters.
In summary, the experimentsshow that the model
C. Hammer properties
givesrealisticsimulationsof the piano stringfor notesin
A key parameterin piano designis the compression different Pitch ranges,at different dynamic levels, and
characteristicof the hammer. For this reason,it was interplayed with differenthammers.Theseresultsare in agreeestingto verify that the numericalmodel reproducesthe
ment with thoseobtainedby other authorsusingdifferent
main featuresof a measuredwaveform,usingreasonable modeling
techniques?
Mostofthediscrepancies
whichocestimates
ofhammer
parameters
K andp.• In theprevious cur can be explained by slight mismatchesin string-
wherea is the relativehammerstrikingposition.
With the parameterslistedin Table I for noteC4, Eq.
(1) predicts,for example,a frequencyshift of nearly400
Hz for partialsnear 5 kHz, which is in closeagreement
with the spectrapresentedin Fig. 5. Other discrepancies
still remain, especiallyfor the highestpartials in theforte
case,which cannotbe explainedwith our model.
1634
J. Acoust. Soc. Am., Vol. 95, No. 3, March 1994
A. Chaigne and A. Askenfelt:Simulationsof piano strings 1634
60
2
m/s
4•
o
:30
-1
0
-2
--
0
•
4
6
6
0
10
4
8
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16
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16
õ
12
16
kHz
NORMAL
•
0
2
4
6
8
10
kHz
6o
dB
30
0
15
0
2
4
6
8
4
10
kHz
FIG. 7. Spectralenvelopescorresponding
to the initial 32 ms of the
simulatedstringvelocitiesin Fig. 6 with threedifferenthammers(C7H-
FIG. 9. Influenceof stringstiffness.
Simulatedstringvelocitiesof a bass
note (C2) for threecaseswith increasing
stiffness;
soft (--$0%), nominal, and stiff (+:50%). The value of the stiffnessparameter was
•= 5.0X 10-6, 7.5X 10-6, and11.25
X 10 6,respectively.
C4H-C2H) strikingnoteC4 (dashed).Themeasured
spectral
envelopes
areincluded
forcomparison
(full lines).
4
II. SYSTEMATIC
VARIATIONS
STRING PARAMETERS
OF HAMMER
AND
hammerparameters,
and by the limitedspatialresolution.
The comparisonbetweenspectraof real and simulated
In the previoussection,the goal was to assessthe
power of our numericalmodel in reproducingthe main
tones show, however, additional differenceswhich cannot
featuresof real pianostrings.It was shown,in particular,
be simplyexplainedwith the helpof our model,especially how the model was able to account successfullyfor the
for frequencies
above3 kHz. At present,it cannotbe con- influenceof bothinitial hammervelocityand felt compreseludedwhetherthey are due to small errorsin the measion properties.These simulationsconfirm the basic
surements,
or to insufficientmodelingof the system.Since sumptionson the behaviorof pianostringsdevelopedin
thesediscrepancies
primarilyreferto spectralcomponents thepastfewyears
byseveral
authors?
whosemagnitudeare at least30 dB belowthe fundamenIn this section,our modelis usedas a tool for systemtal, we couldstill beratherconfidentin usin•g
the modelfor
aticallyexploringthe influenceof somehammerand string
predictions.
parameterson string waveformsand spectra.The param-
20
o
o
I
ms
2
• 20
eters selectedfor the numerical experimentsincluded
string stiffness,hammer-stringmass ratio, and relative
hammerstrikingposition("strikingratio"). Theseparametersare relativelydifficultto changeexperimentally,so a
numericalmethodlendsitselfparticularlywell for thispurpose,oncethe methodhas beenverifiedto performsatisfactorily.The generalapproachusedherewasto startfrom
the measuredvaluesof the parameters(givenin Table I),
and proceedby varying one parameterat a time, stepby
step.
.•
A. String stiffness
0
m
o
I
ms
2
0
1
ms
•
20
o
FIG. 8. Simulatedhammerforcescorresponding
to Fig. 6 with three
differenthammers(C7H-C4H-C2H) strikingnoteCA;.
1635 d.Acoust.
Soc.Am.,Vol.95, No.3, March1994
In Fig. 9, the stringvelocitywaveformof a bassnote
(C2) is shownas the stringstiffnessis variedby +50%
comparedto its nominal value (seeTable I). It is seenthat
the majoreffectof increasing
the stiffness
is to enhancethe
transverseprecursor preceding the reflection from the
bridge. The exampleis a nice illustrationof the ability of
our numericalmodelto reproducedispersive
effectsdueto
stiffness
directlyin the time domain.The interesting
questionof the perceptual
importance
of the transverse
precurA. Chaigne
andA. Askenfelt:
Simulations
of pianostrings 163S
20
sorhasnot beenstudiedasyet, but it mightbe smallerthan
expected.In particular,at the very beginningof the note,
whereonecouldexpecta weak,but early,componentto be
audible,the transversal
stringprecursor
is maskedby lon-
•
0
0
gitudinal
string
motion?
m 20
In the spectra,the effectof stiffness
changes
wasless
pronounced,
visibleonly asthe expectedslightincrease
in
the progressive
spacing
of thepartials.Hall andAskenfelt
•
,
2
o
haveshownthat oneeffectof stiffness
is a slightreduction
ofthemagnitude
ofthehigher
components.
8
20
The influenceof string stiffnesson hammerforce is
rather small in comparisonwith other parameters,which
confirms
the resultsobtainedby Hall.3 The simulated
forces show variations of about +0.5%
of the maximum,
(a)
asthestringstiffness
is variedby + 50%, andthe duration
remains the same.
B. Hammer-string mass ratio
The influenceof changingthe hammer-stringmassratio was most clearly reflectedin the interactingforce between hammer and string. Figure 10(a)-(c) show hammer forces for the notes C2, C4 and C7, each note
includingthree casescorresponding
to differenthammerstringmassratios:Halved, nominal,and doubled,respectively (seeTable I). It canbe seenthat a majoreffectof
makingthe hammerheavieris to increasethe contactduration, and vice versa. For the notes C4 and C7, the increaseis roughly +30% for a doublingand halvingin
hammer-string
massratio, respectively.
For the bassnote
(C2), the resultsare not symmetrical.The changein the
contactdurationfor a halvingin hammer-string
massratio
(b)
is similar to the two other notes (- 20%), but with dou-
bled hammer massa dear caseof multiple contactsdevel-
ops and the total duration is extendedconsiderably
(+60%).
It is interestingto note that a changein the hammerstringmassratiodoesnot influencethe smoothness
of the
forcewaveformto any appreciable
extent.This is in contrast to the caseswherethe dynamiclevel was changed
(Fig. 4), or whenexchanging
hammers(Fig. 8), in which
the forcewaveformpeakedwith increasing
hammervelocity, and increasingstiffness,respectively.
Here, the main
effectis insteadto increasethe numberof pulsesreturning
fromtheagrafieasthehammer-string
massratioincreases,
a heavierhammerrequiringmore accumulatedimpulses
from the agrafiereflections
in orderto leavethe string.
Naturally, the changesin hammer-stringmassratio
are alsovisiblein the stringwaveform,as well as in the
forceactingon the bridge.The bridgeforcescorresponding
to the C4 examplesin Fig. 10(b) are shownin Fig. 12(a).
It can be seenthat the amplitudeof the bridgeforce increaseswith the hammer mass,accompaniedby an increaseof the width of the pulsesdue to the extended
hammer-string
contact.Interestingly,
the repeatedagrafie
reflections,
more pronounced
in the casewith the heavy
hammerasmentioned,are clearlyvisiblealsoin the bridge
forcewaveform.Thisperiodicity,
whichcorresponds
to the
"missing
partials"in the spectradueto strikingposition,
are further discussedin the next section (see Sec. II C).
1636 J. Acoust.
Soc.Am.,Vol.95, No.3, March1994
(c)
o
2
4
FIG. 10. Influenceof hammer-stringmassratio. Simulatedhammer
forcesfor a (a) bass(C2), (b) mid (C4), and (c) treblenote (C7), for
three casesof hammer-stringmassratio; light (-50%), normal, and
heavy( + 100%). Actual massratiosMH/M s were (C2) 0.07, 0.14, and
0.28; (C4) 0.375,0.75, and 1.5; (C7) 2.36, 4.71, and 9.42.
In the experiments
reportedabove,variationsof the
hammer-stringmassratio were obtainedthrough variations of the hammer massonly. The samevaluesof the
hammer-stringmassratio can, of course,be obtained
throughvariationsof the stringmass,the hammermass
remainingconstant.In that case,it becomesnecessary
to
adjustthetensionof thestringin orderto keepthefundamentalfrequency
constant,
but asa consequence
thechar-
acteristic
impedance
of the stringwill be modified.It is
A. Chaigne
andA. Askenfelt:
Simulations
of pianostrings 1636
1o
30
N
5
C2
o
-,5
C4 LI•G•
-10
o
4
mfl
FIG. 11. Influenceof string and hammer masses.Simulatedhammer
forcesfor a bass(C2) note,for a fixednormalhammer-stringmassratio
M/t/M s (0.14). The threecurvescorrespond
to the nominalcase(full
line), doubling{dashedline), and halving(dottedline) of harrLrner
and
stringmasses,respectively.
interestingto noticein this latter easethat the duration of
hammer-stringcontactdoesn'tdependsignificantlyon the
stringmass.In the exampleshownin Fig. 11 for note C2,
it can be seenthat the contacttime increasesby only 5%,
for a doublingof the stringmass.By contrast,the magnitude of the force pulsesare roughly proportionalto the
stringmasses,
whichis in accordance
with the balanceof
forcesat the strikingpoint.
For thenotesof the instrumentwherethepulsesin the
bridgeforceare clearlyseparated,i.e., in the low and medium range of the keyboard,the main differencesin the
spectrabetween the three caseswith light, normal and
heavyhammers,respectively,
are foundin the upperpartials.An exampleis givenfor the bridgeforceof stringCA
in Fig. 13(a), where the heavy-hammerspectrumhas
dropped10 dB comparedto the light-hammercase.This is
a consequence
of thebroadening
of thestringpulses,which
corresponds
to a lower cutoff frequency.Also, a slightly
weaker fundamental could be observedfor the lighthammer
0
•'
-5
m
õ
8
6
8
0
0
-õ
-10
(a)
o
2
m4s
A
0
• N
0.5
I
ms
A
1.5
A
2
2.5
5
FIG. 12. Influenceof the hammer-stringmassratio. Simulatedforcesat
thebridgefor threecasesof hammer-string
massratio;light,normal,and
heavy.(a) Midrange,C4 [corresponding
to Fig. 10{b)]; (b} treble,C7
[corresponding
to Fig. 10(c)].
case.
In the upper rangeof the keyboard,as illustratedby
the C7 example, bridge forces obtained with different
hammer-stringmassratiosshowan interestingdetail [see
Fig. 12(b)]. Here, the smoothestwaveformis obtained
with the nominalvalue of hammer-stringmassratio, and
not with the highesthammer-stringmassratio like in the
previousexamplefor note C4. For the light hammercase,
the pulsesare narrow enoughto just becomeseparated,
with a hint of zero displacement
in between.In the heavy
hammer case, the waveform becomesmore peaked and
triangular,and the first pulseis clearly higher than the
following,both phenomenabeingthe resultof overlapping
betweenpulses.
In the spectral domain, the C7 case with nominal
hammer-stringmassratio (correspondingto a real instrument) givesthe leastamountof high-frequencyenergy.In
particular,the differencecomparedto the light-hammer
caseis substantialwith a clearlysteeperspectralslopefor
the nominalease,resultingin a 10-dBleveldifferenceat 12
knz [see Fig. 13(b)]. For the two lowest partials, the
heavy-hammercaseshowsa steeperslopein spectrallevel
than for the nominal case,like in Fig. 13(a), but for frequenciesabove4 kHz, a "sawtooth"pattern with more
1637
m4s
'•N•ø
I
NI0]
a,
•: -1o
•
2
J. Acoust.Soc. Am., Vol. 95, No. 3, March 1994
energyin the high-frequency
rangeis observed.
Intermediate valuesof hammer-stringmassratio have alsobeeninvestigated,
showingthat the nominalvaluecorresponds
to
the steepest
slopewith a smoothpattern.
It is an openquestion,whetherthe manufacturers
are
awareof the existenceof this "leasthigh-frequency
ratio"
for the hammerand stringmasses,
and if this is exploited
in the design.All threeexamplesin Fig. 12(b) havebeen
observedin measurements
on real pianos,however,at different partsin the trebledependingon manufacturerand
instrument
size.
C. Striking ratio
The influenceof the relativehammer strikingposition
(strikingratio) wasalsoinvestigated.
Figure 14(a) shows
the changesin waveformanddurationof the hammerforce
for note C4, as the strikingratio was variedbetween0.04
(1/25) and 0.24 (1/4.2). The nominal value for the C4
simulations was 0.12 (= 1/8 ), which is representativefor
the striking ratios for this note in most pianos (between
1/9=0.11 and 1/7..•0.14). As expected,it canbe seenthat
the reflectionsreturningfrom the agrafiebecomemoreand
A. Chaigne and A. Askenfelt:Simulationsof piano strings 1637
60
dB
45
30
15
0
0
(a)
2
4
0.õ
I
1 .,5
2
2.5
(a)
dB
C7
45
3O
15
(b)
5
•o
]5
kHz
20
FIG. 13. Spectralenvelopes
corresponding
to the initial 32 ms of the
simulatedbridge forcesin Fig. 12, showing (a) note C4, and (b) note
C7,with three hammer-string mz•s ratios; light (thin line), normal
(dashed),and heavy(broadline). The actualmassratiosMn/M s are the
sameas in Fig. 10.
o
o
(b)
0.1
RELATIVE
O.P
STRIKING
0.3
POSITION
2O
more separatedand pronounced,as the strikingratio is
increased.For a very short striking ratio (0.04) the force
resemblesa smoothhump, while for the nominalcaseand
C,
for longerstrikingratios,two distinctiveagrafiereflections
O
can be observed in the force waveform. As the release of
the hammerfrom the stringis dependenton the accumulated effectof severalagrafiereflectionsimpingingon the
hammer,
s thecontact
duration
increases
as thestriking
ratio is increased.In the simulations,the relation between
striking ratio and contactduration was found to be essentially linear [seeFig. 14(b)]. For the C4 note,the contact
duration doubledfor a sixfold increasein striking ratio
(0.04 to 0.24).
The changein hammer-stringinteractionwith increas-
ing strikingratio, characterized
by a progressive
divisionof
one broad forcehump into severalseparatedpulses,is illustratedin an alternativeway in Fig. 14(c). This figure
showshow the maximum value of the force pulsedecreases
as the strikingratio is increased,meaningthat, for small
relative hammer striking positions,the accumulationof
agrafiereflectionsdoesn'tallow the force to decayin between.
The influenceof the strikingratio on the forcewaveform at the bridge,and on the corresponding
spectra,was
studiedfor a bassnote (C2), by halvingand doublingthe
nominalstrikingratio (seeFig. 15). The well-definedpattern for a shortstrikingratio, with pulsesat the fundamental period spacing,is successively
more and more "disturbed" by pulsesin between,as the striking ratio is
increased.These interferingpulsescorrespondto the in1638 J. Acoust.Soc. Am., Vol. 05, No. 3, March 1994
0
(C)
0.1
0.2
0.3
RELATIVESTRIKINGPOSITION
FIO. 14. Influenceof relativehammerstrikingpositiona for noteC4. (a)
Simulated hammer forces for cz between 0.04 and 0.24, (b) hammer-
stringcontactdurationversusa, (c) maximumhammerforce versusa.
creasinglydelayedfirst and secondagrafiereflections(indicated by A 1 and A2).
In the frequencydomain,the effectof increasingthe
strikingratio is seenmainly in the periodicalpatternpunctuatingthe spectralenvelopewith weakmodescorresponding to the inverseof the strikingratio (see Fig. 16). The
repetitionrate of the agrafie reflections,interfoliatingthe
fundamentalpulses,correspond
to these"missingmodes"frequencies,
but they are not strongenoughto fill up the
gapsin the spectra,whenthey are "givento" the stringat
theendofthestring
contact.
m4'l•
Theoverall
spectral
slope
doesnot changeas the strikingratio is varied.
A. Chaigneand A. Askenfelt:Simulationsof piano strings 1638
10
N
5
0
-
-5
-10
o
•
ms
20
30
10
ms
20
30
10
ms
20
30
o
ea -10
o
10
N
6
0
-
-6
-10
0
FIG. 15. Influenceof strikingpositionfor noteC2. Simulatedforceat the
bridgefor threecasesof increasing
relativehammerstrikingposition;
short( - 50%), normal,and long ( + 100%). The waluefor the relative
strikingpositionwas =0.06, 0.12, and 0.24, respectively.
The first and
secondreflections
from the agrafieare indicatedby A1 and A2, respectively.
cordedby appropriatestringtransducers.The influenceof
the frequency-dependent
damping,whichis relativelyhard
to seein the waveforms,is clearly audible.
The flexibilityof the simulationprogrammakesit possibleto usethe modelasa efficienttool for exploringa wide
range of physical parameters,such as string stiffness,
hammer-stringmassratio, and relativehammerstriking
position.Sucha systematic
explorationof the pianodesign,
guidedby a perceptualevaluation,is probablyone of the
major featuresof soundsynthesis
by physicalmodeling.
Our numericalmodel of the piano string is still far
from complete.Amongotherthings,the boundaryconditionsare modeledin a rathercrudeway. Work in progress
include an attempt to terminatethe last segmentof the
string by a realisticload, obtainedfrom admittancemeasurementsat the bridgeof a real piano (at the pointwhere
the actualstringcrosses).Pilot experiments
havealready
beenconductedwith simulationsof a triplet of slightly
detunedC6 stringscoupledby the bridge,usingexisting
measurements
of thesoundboard
admittance.
i?Thisapproachhasgivensignificantimprovements
in the realismof
the synthesis.
By includinga modelingof the two directionsof stringpolarization,onein parallelandoneperpendicular to the soundboard,we expectto obtain a more
realisticdecayprocess,
dueto the energyexchangebetween
thetwofamilies
of modes.
is For thispurpose,
somepre-
III. CONCLUSIONS
The evaluation of our numerical model has shown that
it is capableof reproducingthe characteristicfeaturesof
real pianostringsin boththe timeand frequency
domain.
The influenceof the controlparametersfor the excitation
of the struckstring(suchas initial hammervelocityand
felt compressionproperties), discussedby several
authors,
3'?'•6
hasbeenconfirmed.
In addition,
informal
listening testshave shown that the synthesizedtonessound
very similar to the vibrationsof real piano strings,as re6O
liminary measurementsof the admittancematrix at the
bridgewill be necessary.
Presently,it must be admitted that the simulatedtones
don't mimic real pianotonesconvincingly
whenlistening,
a slightlydisappointing
fact. The essentialmissingfeature
in the synthesis
is in the attackcomponent,whichfor a real
pianotoneincludesa strong"thump."This component
is
due to the shocklike excitation
of the whole instrument
at
the impactof the hammeron the string.A largepart of the
thump originatesfrom the soundboard,which is set in motion almostimmediatelyat the hammerimpactby a longitudinally transmitted pulse train reflecting between
bridgeandhammer.This longitudinalmotionprecedes
the
firsttransversal
stringpulse1-2ms(midrange).
• A good
0
•
6o
•
•o
e•dB .•
•
'•'
I
2
3
4
kltz
5
illustrationof this effectcan be obtainedby isolatingthe
stringtransmittedthumpfrom the "sounding"part of the
tone by dampingthe transversalstring motion (with the
hand).
, .It•
•..•
[NOI•
A•L•
0
Other transientcomponentsare excitedby the inertial
forceof the acceleratinghammer,as well asthe touchforce
on the key, which both are transmittedto the iron plate
I
2
3
4
5
dB
45
and soundboard
via the key bedand rim. An exampleof
the attack components,as registeredat the bridge,
shownin Fig. 17, for a casewherethe stringshavebeen
removedand the hammerstrikesa dummy mass.This was
I
2
3
4
kHz
5
FIG. 16. Spectraof the simulatedbridgeforcesin Fig. 15 for note C2
with threerelativehammerstrikingpositions;
short,a=0.06 (1/16.7),
normal,a=0.12 (1/8.3), andlong,a=0.24 (1/4.2). The dashedvertical
linesindicatethefirstminimadueto therelativestrikingposition.
1639 J. Acoust.
Soc.Am.,Vol.95, No.3, March1994
donein orderto avoida maskingof theattackcomponents
by the much strongerstring vibrations.Interestingly,:it
appearsthat the particularhistoryof the attackcomponentsare affectedby the pianist'stype of touch.
By addinga recordingof the attack components(with
dampedstrings) to the computersimulations,significant
improvementsin the realism of the synthesiswere obtained.Thishasencouraged
usto continuethemodeling
of
A. Chaigne
andA. Askenfelt:
Simulations
of pianostrings 1639
5 mS/DIV
tion. The authors wish to thank Xavier Boutilion and an
anonymousreviewerfor helpfulcommentson a previous
'• ' ' KEY
BOTTOM
versionof this paper.
•A. Chaigne
andA. Askenfelt,
"Numerical
simulation
of pianostrings.
HAMMER
-DUMMY•
I: A physicalmodelfor a struckstringusingfinitedifference
methods,"
J. Acoust. Soc. Am. 95, 1112-1118 (1994).
2H. SuzukiandI. Nakamura,"Acoustics
of pianos,"Appl.Acoust.30,
147-205 (1990).
'
v•v
3D. Hall, "Pianostringexcitation.
VI: Nonlinear
modeling,"
J. Acoust.
Soc. Am. 92, 95-105 (1992).
4A. AskenfeltandE. Jansson,
"Fromtouchto stringvibrations--The
initial courseof the pianotone,"SpeechTransmission
Lab. Quarterly
Progress
andStatusReport,Dept.of SpeechCommunication
andMusicAcoustics,
RoyalInstituteof Technology,
Stockholm,STL-QPSR1,
FIG. 17. Registration
of the acceleration
at the bridgewith the strings
removedand the hammerstrikinga dummymass(C4, forte, staccatotouch).The signalsrepresenting
hammer-dummy
contact,andkey bottom position,are includedfor reference
(horizontalbars).The bridge
motionstartswith a percussive
component
about20 msbeforehammerdummycontact,followedby a "wavepackage"at about 1000Hz (due to
a resonance
in the key). Later, the motionis dominatedby a mixtureof
resonances
in the keybedat 100and250 Hz, approximately.
The vibrationlevelof thesepre-andpostcursors
is of thesameorderasthebridge
variationsdueto stringmotionat pp level.
31-109 (1988).
SA.Askenfelt
andE. V. Jansson,
"Fromtouchto stringvibrations.
III:
"Stringmotionand spectra,"J. Acoust.Soc.Am. 93, 2181-2196
(1993).
6j. Nakamura,"Thevibration
of a struckstring(Acoustical
research
on
the piano.Part 1)," J. Aeoust.Soc.Jpn.(E) 13(5), 311-321(1992).
7X. Boutilion,"Modelfor pianohammers:
Experimental
determination
and digitalsimulation,"J. Acoust.Soe.Am. 83, 746-754 (1988).
SD.Hall andA. Askenfelt,
"Pianostringexcitation.
V. Spectra
for real
hammersandstrings,"J. Acoust.Soc.Am. 83, 1627-1638(1988).
9D. Hall,"Pianostringexcitation
in thecaseof a smallhammer
mass,"
J. Acoust. Soc. Am. 79, 141-147 (1986).
tOT.Yanagisawa
andK. Nakamura,
"Dynamic
compression
characteris-
the longitudinalvibrationsof the string,oncethe soundboardadmittanceat the bridgeterminationhavebeensatisfactorilyimplemented.
ticsof pianofelt," J. Acoust.Soc.Jpn.40, 725-729 (1984) (in Japanese).
n A. Askenfelt
andE. V. Jansson,
"Fromtouchto stringvibrations.
II:
"The motionof the key and hammer,"J. Acoust.Soc.Am. 90, 23832393 (1991).
ACKNOWLEDGMENTS
Part of this work was conducted during Fall 1990,
whenthe firstauthorwasa guestresearcher
at the Department of SpeechCommunicationand Music Acoustics,
Royal Instituteof Technology(KTH), Stockholm,with
financialsupportfrom Centre National de la Recherche
Scientifique(CNRS). The projectwas further supported
by the SwedishNatural ScienceResearchCouncil(NFR),
the Swedish Council for Research in the Humanities and
Social Sciences(HSFR), the Bank of Sweden Tercente-
nary Foundation,and the Wenner-GrenCenter Founda-
1640 J. Acoust.
Soc.Am.,VoL95, No.3, March1994
12D. Hall,"Pianostringexcitation.
II: General
solution
fora hardnarrow
hammer," J. Acoust. Soc. Am. 81, 535-546 (1987).
•3M. Podlesak
and A. Lee, "Dispersion
of wavesin pianostrings,"J.
Acoust. Soc. Am. 83, 305-317 (1988).
14A. H. Benade,
Fundamentals
of MusicalAcoustics
(Oxford,U.P.,New
York, 1976), Chap. 17.
•SD.Hall and P. Clark,"Pianostringexcitation.
IV: The question
of
missingmodes,"J. Acoust.Soc.Am. 82, 1913-1918 (1987).
•6H. Suzuki,"Modelanalysis
ofa hammer-string
interaction,"
J. Acoust.
Soc. Am. 82, 1145-1151 (1987).
i?K. Wogram,
"Acoustical
research
onpianos:
Vibrational
characteristics
of the soundboard,"Das Musikinstrument 24, 694-702, 776-782, 872880 (1980).
tSG.Weinreich,
"Coupled
pianostrings,"
J. Acoust.Soc.Am.62, 14741484 (1977).
A. Chaigne
andA. Askenfelt:
Simulations
of pianostrings 1640